Determination of excess chemical potentials

If we take the second and third components as those that form the mixed solvent and express the ratio x2/x3 as k, a constant, then we can eliminate x2 from Equation (10.201), so 

Furthermore, if we choose to express the composition of the system in terms 

The quantity ps can be considered as the chemical potential of the mixed solvent. The Gibbs-Duhem equation may be written as 

Equations (10.205) and (10.207), together with Equation (10.206), comprise the basic equations to use in considering a ternary system that has a constant ratio of x2/x3 as a pseudobinary system. 

The study of a ternary system over the entire range of composition presents a formidable experimental problem. Several methods have been developed by which the excess chemical potentials of two of the components can be calculated from known values of only one component. Only two of these methods are discussed here. Throughout the discussion we assume that the values of the excess chemical potential of the first component are measured or known at constant temperature and pressure. 

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Say you have performed a classical calculation to determine the excess chemical potential from the first two terms on the right side of (11.22) followed by another classical calculation to obtain an estimate of the quantum correction from the expression (11.29), and the estimated correction is large. This suggests that a full quantum treatment is necessary. In this section, we derive the appropriate formulas for changes in the excess chemical potential due to mutating masses. If the original mass is very large, which corresponds to the classical limit, the derived expressions yield the quantum correction. 

Nagasawa and Takahashi, 1972). More specifically, the value of the second virial coefficient determines the excess chemical potential, juE (also known as the excess partial molar Gibbs free energy), which characterizes the formation of biopolymer-solvent and biopolymer-biopolymer pair contacts ... 

The experimental studies of three-component systems based on phase equilibria follow the same principles and methods discussed for two-component systems. The integral form of the equations remains the same. The added complexity is the additional composition variable the excess chemical potentials become functions of two composition variables, rather than one. Because of the similarity, only those topics that are pertinent to ternary systems are discussed in this section of the chapter. We introduce pseudobinary systems, discuss methods of determining the excess chemical potentials of two of the components from the experimental determination of the excess chemical potential of the third component, apply the set of Gibbs-Duhem equations to only one type of phase equilibria in order to illustrate additional problems that occur in the use of these equations, and finally discuss one additional type of phase equilibria. 

Here, we report some basic results that are necessary for further developments in this presentation. The merging process of a test particle is based on the concept of cavity function (first adopted to interpret the pair correlation function of a hard-sphere system [75]), and on the potential distribution theorem (PDT) used to determine the excess chemical potential of uniform and nonuniform fluids [73, 74]. The obtaining of the PDT is done with the test-particle method for nonuniform systems assuming that the presence of a test particle is equivalent to placing the fluid in an external field [36]. 

Excess chemical potentials As early as 2002 Lynden-Bell et al. published an investigation about the chemical potential of water and organic solutes in [C,mim] [Cl] [9], The authors stated ... the chemical potential is the most important thermodynamic property of a solute in solution because it determines the solubility and chemical reactivity of a solute. Within this seminal article the authors determined the excess chemical potential (pf) by means of theoretical methods. The excess chemical potentials pA of a series of molecules dissolved in the IL [C,mim][Cl] were calculated by a sequence of transformations [9], It is defined by... 

If a segment has a zero Rosenbluth weight then growth of the chain is terminated. However, such chains must stiU be included in the averaging used to determine the excess chemical potential. 

In order completely to determine the excess chemical potential, the activity coefficient, or the activity, the convention used to define the reference value of the chemical potential and the composition units used, must be specified. 

Many experimental techniques are used to determine the values of excess chemical potentials or activity coefficients. In this section we briefly discuss the relationships between the activity coefficients and information derived from measurement of... 

On the basis of these assumptions and by integration of Poisson s equation with an approximate charge density, the potential at any distance from a given ion surrounded by an atmosphere of other ions is obtained. This potential is used together with a charging process on the ions to determine the excess chemical potential due to the electrostatic interactions. A major result of the theory is the determination of expression for single-ion activity coefficients in the form... 

The results for the chemical potential determination are collected in Table 1 [172]. The nonreactive parts of the system contain a single-component hard-sphere fluid and the excess chemical potential is evaluated by using the test particle method. Evidently, the quantity should agree well with the value from the Carnahan-Starling equation of state [113]... 

Figure 5 shows pn distributions for spherical observation volumes calculated from computer simulations of SPC water. For the range of solute sizes studied, the In pn values are found to be closely parabolic in n. This result would be predicted from the flat default model, as shown in Figure 5 with the corresponding results. The corresponding excess chemical potentials of hydration of those solutes, calculated using Eq. (7), are shown in Figure 6. As expected, /x x increases with increasing cavity radius. The agreement between IT predictions and computer simulation results is excellent over the entire range d < 0.36 nm that is accessible to direct determinations of po from simulation.

At equilibrium, the chemical potential for a given molecular species is constant throughout the system. The two terms on the right-hand side of (11.4) can vary in space, however, so as to add up to a constant. In an inhomogeneous system, the number density and excess chemical potential adjust so as to yield the same constant chemical potential. Due to the local nature of the excess chemical potential, it is reasonable to define an excess chemical potential at a single point in space and/or for a single molecular conformation [29]. That excess chemical potential then determines... 

In this chapter, we shall consider the methods by which values of partial molar quantities and excess molar quantities can be obtained from experimental data. Most of the methods are applicable to any thermodynamic property J, but special emphasis will be placed on the partial molar volume and the partial molar enthalpy, which are needed to determine the pressure and temperature coefficients of the chemical potential, and on the excess molar volume and the excess molar enthalpy, which are needed to determine the pressure and temperature coefficients of the excess Gibbs function. Furthermore, the volume is tangible and easy to visualize hence, it serves well in an initial exposition of partial molar quantities and excess molar quantities. 

Using umbrella sampling, Tieleman and Marrink [18] determined a PMF for transferring a DPPC lipid from water to the center of a DPPC bilayer (Figure 3B). The DPPC PMF has a deep minimum at its equilibrium position and a steep slope in free energy as it moved into bulk water. The free energy of desorption (AGdesorb) was 80 kj/mol, and is directly related to its excess chemical potential in the bilayer compared to water. 

The reference state of each component in a system may be defined in many other ways. As an example, we may choose the reference state of each component to be that at some composition with the condition that the composition of the reference state is the same at all temperatures and pressures of interest. For convenience and simplicity, we may choose a single solution of fixed composition to be the reference state for all components, and designate xf to be the mole fraction of the /cth component in this solution. If (Afikx) represents the values of the excess chemical potential based on this reference state, then (A/if x ) [T, P, x ] is zero at all temperatures and pressures at the composition of the reference state. That this definition determines the standard state is seen from Equation (8.71), for then... 

One-component, two-phase systems are discussed in the first part of this chapter. The major part of the chapter deals with two-component systems with emphasis on the colligative properties of solutions and on the determination of the excess chemical potentials of the components in the solution. In the last part of the chapter three-component systems are discussed briefly. 

Three different uses of the Gibbs-Duhem equation associated with the integral method are discussed in this section (A) the calculation of the excess chemical potential of one component when that of the other component is known (B) the determination of the minimum number of intensive variables that must be measured in a study of isothermal vapor-liquid equilibria and the calculation of the values of other variables and (C) the study of the thermodynamic consistency of the data when the data are redundant. 

A) When only one of the two components in a binary solution is volatile, the excess chemical potential of the volatile component can be determined by the methods that have been discussed. However, we require the values of the excess chemical potential of the other component or of the molar... 

It is apparent from this equation that we can determine only the difference between the two excess chemical potentials at the experimental conditions. If one is known from other studies, then the other can be determined. It must be pointed out that the difference is not isothermal and, if isothermal quantities are desired, the same corrections as discussed in Section 10.12 must be made by the use of Equation (10.87). 

The measurement of osmotic pressure and the determination of the excess chemical potential of a component by means of such measurements is representative of a system in which certain restrictions are applied. In this case the system is separated into two parts by means of a diathermic, rigid membrane that is permeable to only one of the components. For the purpose of discussion we consider the case in which the pure solvent is one phase and a binary solution is the other phase. The membrane is permeable only to the solvent. When a solute is added to a solvent at constant temperature and pressure, the chemical potential of the solvent is decreased. The pure solvent would then diffuse into such a solution when the two phases are separated by the semipermeable membrane but are at the same temperature and pressure. The chemical potential of the solvent in the solution can be... 

Values of Apf are determined similarly by the use of Equation (10.220) expressed in terms of the excess chemical potentials. In this case the new variables would be... 

Once the species present in a solution have been chosen and the values of the various equilibrium constants have been determined to give the best fit to the experimental data, other thermodynamic quantities can be evaluated by use of the usual relations. Thus, the excess molar Gibbs energies can be calculated when the values of the excess chemical potentials have been determined. The molar change of enthalpy on mixing and excess molar entropy can be calculated by the appropriate differentiation of the excess Gibbs energy with respect to temperature. These functions depend upon the temperature dependence of the equilibrium constants. 

This equation requires a knowledge of both xAB and xAB in order to determine the difference between the excess chemical potentials. The quantity AfiAB cannot be evaluated generally, and it is the difference (AfiAB — AfFAB) that is determined experimentally. 

One of the most challenging tasks in the theory of liquids is the evaluation of the excess entropy Sex, which is representative of the number of accessible configurations to a system. It is well known that related entropic quantities play a crucial role, not only in the description of phase transitions, but also in the relation between the thermodynamic properties and dynamics. In this context, the prediction of Sex and related quantities, such as the residual multiparticle entropy in terms of correlation functions, free of any thermodynamic integration (means direct predictive evaluation), is of primary importance. In evaluating entropic properties, the key quantity to be determined is the excess chemical potential (3pex. Calculation of ppex is not straightforward and requires a special analysis. 

The advantage of this choice of the X dependence for the correlation functions and the bridge function relies on the fact that the excess chemical potential, and the one-particle bridge function as well, can be determined unambiguously in terms of B(r) as soon as n and m are known. To address this problem, the authors proposed to determine the couple of parameters (n m) in using the Gibbs-Duhem relation. This amounts to obtaining values of n and m from Eq. (87), which is considered as supplementary thermodynamic consistency condition that have to be fulfiled. 

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